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On the real exponential field with restricted analytic functions. (English) Zbl 0823.03017
The present paper generalizes a result of A. Wilkie who has proved the model-completeness of the structure $$\mathbb{R}_{\exp}$$, i.e. the field of reals together with exponentiation. In an earlier paper, Denef and van den Dries have already proved the model-completeness of $$\mathbb{R}_{\text{an}}$$, the field of reals endowed for every $$m\geq 1$$ with the restriction to $$[- 1,+ 1]^ m$$ of all $$m$$-ary real-valued functions $$f$$, analytic in a neighbourhood of $$[- 1, +1]^ m$$. Now, this paper gives a proof of the model-completeness of the structure $$(\mathbb{R}_{\text{an}},\exp)$$, i.e. $$\mathbb{R}_{\text{an}}$$ together with real exponentiation. The proof generalizes that of A. Wilkie and, in addition, gives o-minimality for the structure $$(\mathbb{R}_{\text{an}}, \exp)$$. There is a more recent work of the first author, A.Macintyre and D. Marker [Ann. Math., II. Ser. 140, 183-205 (1994)] which proves an even stronger result for $$(\mathbb{R}_{\text{an}}, \exp)$$ by using a different method in the proof.

##### MSC:
 03C60 Model-theoretic algebra 12L12 Model theory of fields
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##### References:
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